We propose a natural and simple model for the joint throughput evolution of a set of TCP sessions sharing a common bottleneck router, via products of random matrices. Surprisingly enough, even when based on the assumption of i.i.d losses, this model is sufficient to explain the fractal scaling of aggregates of TCP traffic at short time scales, which was recently identified by a statistical analysis based on wavelets by Feldmann, Gilbert, Willinger et al. in [8,7]. This model has various other implications, also described in the paper, such as the quantification of instantaneous fairness, the characterization of the autocorrelation function of inter-loss periods, or the evaluation of the performance degradation due to synchronizati- on of losses in the shared bottleneck router.